Pseudorotations of the -disc and Reeb flows on the -sphere

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

Nonlinear flag manifolds as coadjoint orbits

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

From Pseudo-Rotations to Holomorphic Curves via Quantum Steenrod Squares

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element and hence nontrivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on nonvanishing Gromov–Witten invariants of manifolds admitting pseudo-rotations.

Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Given a symplectic surface [Formula: see text] of genus [Formula: see text], we show that the free group with two generators embeds into every asymptotic cone of [Formula: see text], where [Formula: see text] is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.

Disjoint superheavy subsets and fragmentation norms

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on [Formula: see text] some of them are [Formula: see text]-continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi.

Calabi quasimorphisms for monotone coadjoint orbits

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.